Existence and non-existence of solutions for a class of Monge-Ampère equations

Abstract

In this paper, we study the boundary value problems for a class of Monge-Ampère equations: $det{D}^{2}u=e^{-u}$ in $\Omega \subset {ℝ}^n$, $n\ge 1$, $u{|}_{\partial \Omega }=0$. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we reduce the equation to an ODE, and show there exists a critical radius such that if the radius of a ball is smaller than this critical value, then there exists a solution and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. We calculate the one dimensional case explicitly, which also indicates some kind of bifurcation phenomena may exist. Finally for the fixed domain we consider an equivalent problem with a parameter $det{D}^{2}u=e^{-tu}$ in $\Omega$, $u{|}_{\partial \Omega }=0$, $t\ge 0$. By using Lyapunov-Schmidt Reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimates and bifurcation theory we get the global structure and prove existence of at least two solutions for a certain range of parameters $t\in \left(0,T^{*}\right)$, a unique solution at $t=T^{*}$, nonexistence of solution for $t>T^{*}$.